Abstract
An important point of contention in the philosophy of mathematics concerns the existence of mathematical objects. Platonists believe they exist independently; nominalists, that they are only linguistic constructs; formalists, that mathematics is not at all a science of objects. I believe the existence of mathematical objects is in fact immaterial for the understanding of the nature of mathematical knowledge. Mathematical truths are formal and only the formal properties of arbitrary domains of objects – whether they exist on their own or are only “intentional correlates” of their theories – matter to mathematics. This perspective has the advantage of making the applicability of mathematics in science less “unreasonable”, connecting it directly to the indifference of formal truth to material context. In this paper I intend to argue for an epistemologically relevant ontologically uncommitted formalist philosophy of mathematics (far from the “rules of the game” variety of formalism) that strips the ontological problem of its philosophical relevance and renders the applicability problem more treatable.
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Notes
- 1.
To be honest, Benacerraf, who brought causal considerations into the philosophy of mathematics (cf., for instance, “Mathematical Truth”, in Benacerraf and Putnam, 1983), seems to be talking of causality in a rather loose way, as some sort of “connection” between the reasons for our belief in the truth of a mathematical assertion and the reasons for it being true. But in general, in mathematics, the reason for our belief in the truth of a mathematical proposition lies in the fact that this proposition follows by acceptable arguments from either conventional stipulations or “intuitive” truths, which coincides with the reason for it being true. The reason for an empirical assertion to be true, on the other hand, is the state of the world, and so the state of the world must, in causal theories of knowledge, be connected in a relevant way to our belief in the truth of empirical assertions. These situations are widely different; why should epistemology treat them uniformly? Such uniformity seems only desirable for those who believe in the existence of an independent world of mathematical entities, on a par with the empirical world, to where we can ascend by means of a form of perception called, faute de mieux, “intuition”; that is, for those leaning towards empiricism and naturalism. (Nonetheless, I believe we can still make sense of the notion of mathematical intuition, but construed in an entirely different fashion.)
- 2.
Aristotelism in the philosophy of mathematics claims that mathematics is only the study of some abstract aspects of empirical reality. Even if developed along these lines a causal theory of mathematical knowledge would face difficulties explaining how we can causally interact with abstract aspects of reality. Moreover, although we can argue that some mathematical domains are abstracted from our experience (provided we have a good theory of abstraction) the vast majority of them are obviously not. Analytically inspired Aristotelism is particularly troubled by the problem of abstraction. How to handle it after Frege gave it such a bad name? The way out is usually dropping Aristotelism in favor of full-fledged Platonism (which accords mathematical objects an existence just like that of real objects, but in a realm out of this world) or nominalism (symbols are, after all, real entities).
- 3.
The idea that they may be merely intentional objects – see note 5 – is not a viable alternative in analytic circles (probably because this smacks so much of psychologism to analytic sensibility).
- 4.
Logical formalization, although important in metamathematics, does not play a relevant role in mathematics. The extensive use of formal-logical arguments in the philosophy of mathematics then risk loosing mathematics as practiced by mathematicians from sight, substituting it with reconstructions that are almost never, if ever, adequate. If we want to understand arithmetic, for instance, its heuristic methods, its ways of validation, the nature of the knowledge it provides, it would be misleading to consider only formal versions of arithmetic, no matter in which logical context.
- 5.
There is a way in which we can understand mathematical existence which lies somewhere between naturalistic inspired Platonisms and psychologism (which is also, of course, a naturalistic perspective). Analytic philosophers tend to confuse it with psychologism, but it is essentially different from it. I am referring to intentional existence. Objects of a certain type (for instance, numbers) exist intentionally to the extent that they are posited, or presupposed, by a theory (in our example, arithmetic), and as long as this theory maintains its logical consistency. Intentional existence of objects is then parasitic on the existence of a logically coherent theory of these objects. The objectivity of a theory, i.e. the fact that it is shared by an entire community (the mathematical community in our example) is inherited by the objects the theory posits – numbers, in our case, are objective entities for the community of arithmeticians to the extent that this community agrees that they are talking about “the same thing” when they are doing arithmetic. The moment a theory manifests an inconsistency their objects, in the word of Husserl, “vanish”. Mathematical existence is then closely tied to logical consistency, just as Hilbert and Poincaré, among many, wanted. I understand that Frege is not far from this perspective. The so-called context principle, after all, tells us not to ask for the meaning of a term outside a context in which it occurs. Numbers are, for Frege, objectively existing logical objects to the exact measure that they occur as referents of numerical terms in the context of what Frege took for a logical theory, arithmetic. The hypostasis of mathematical objects occurs when intentional existence is taken for theory-independent and self-subsistent existence (the adoption of a naturalistic inspired correspondence theory of truth goes in general hand in hand with this).
- 6.
By this I mean that mathematical theories and the structures, forms or formal domains they characterize are in general invented by creative mathematical minds rather than imposed by pre-existing “mathematical facts”. In fact, not even Euclidian geometry can be said to simply describe our experience of physical space or our intuition of pure space, as Kant believed (see Helmholtz 1866, 1870). Our experience of space is too coarse to impose any geometry, and there is no pure intuition of space as Kant believed.
- 7.
The expression is due to Wigner (Wigner, 1960), who thought there was something mysterious and inexplicable in the mundane fact that mathematics is useful in physics (“a wonderful gift which we neither understand nor deserve”). Wigner was the first to raise seriously the question of how to account for the “miracle” of the “appropriateness of the language of mathematics for the formulation of the laws of physics”. For him, mathematics is to a large extent done for aesthetic reasons, and the fact that Nature favors the language of mathematics is a wonder we do not understand (nor deserve). The number of times the word “miracle” is used in his article already indicates the frame of mind with which he approaches the problem. Steiner will later stress this mystic undertone.
- 8.
Steiner (Steiner, 1998) believes that, on top of offering a convenient language and a conceptual apparatus for science, mathematics can also play a heuristic role in it. More specifically, he thinks that purely mechanical manipulations of symbols can lead to findings in physics. One of his favorite examples is Maxwell’s discovery of electromagnetic waves (later experimentally confirmed by Hertz). According to Steiner (see Steiner, 1989, p. 458), Maxwell realized that the equations of electromagnetism he received from his predecessors were inconsistent with the preservation of electric charge. Then, by playing with these equations, Steiner says, Maxwell hit on the notion of displacement current, its mathematical expression and the hypothesis that displacement currents also generate magnetic fields. The stage was then set for the discovery of electromagnetic waves. My first reaction to this account is the obvious one: even if it were historically accurate (which it isn’t), Maxwell would not be only playing with mathematical symbols, but working out the mathematical consequences of a physical hypothesis, namely, that electric charge must be conserved, the truth, however, is that the concept of displacement current and its mathematical expression were natural outcomes of the physical model for electromagnetic phenomena Maxwell worked with (mechanical stresses and displacements in an elastic medium transferred from one point to another in finite time by contact).
- 9.
For a detailed discussion of this argument see Colyvan 2001b.
- 10.
To think of mathematical proofs as formal proofs is a heritage of the logicist approach to the philosophy of mathematics. But logicism is a reinterpretation of mathematical practice, devised for strictly foundational goals, not an unbiased view of what proving in mathematics is all about.
- 11.
Of course, I have in mind Benacerraf’s famous example (see Benacerraf, 1965). The set-theoretical translation of a mathematical theory should not be understood as an ontological reduction – as if this theory were really about certain types of sets –, but only as a different materialization of a collection of formal truths. The fact that mathematical concepts can be translated into set-theoretical concepts does not give sets any privileged ontological status.
- 12.
Strictly speaking, this is to a large extent precisely what real numbers are. It is doubtful we can have any clear intuition of any definite real number, even in the form of a definite ratio between two segments (our perceptual or intuitive powers would not be able to discern it from another ratio differing only slightly from it). A real number is an idealization, a product of the imagination, not anything “given” to us. The domain of all real numbers, i.e., the system of all conceivable such ratios abstractly considered, is even more obviously a scheme of understanding, which cannot in any clear sense be intuited or perceived. In general, mathematical theories, such as geometry, group theory, set theory, or arithmetic, may be suggested by our experience (experience can at beast trigger mathematical imagination), but in the end they are never mere descriptions of anything we simply “experience”.
- 13.
This quote occurs in a letter to Carl Stumpf of 1890 or 1891 (Willard, 1994, pp. 12–19). Husserl’s own answer to this question of paramount importance, although ingenious, is unsatisfactory. For him, a purely formal consistent extension of a theory, written in a richer language, can be used for deriving results in this theory provided it can do so, but in an inessential way; i.e. provided the theory it extends is logically complete (with respect to its own language). The answer is not satisfactory because it is unnecessarily restrictive, as I will show.
- 14.
For Husserl, a formal domain (or formal manifold) is the “objective correlate” of a purely formal theory. In less threatening words, a formal domain is what we get by stipulating (a sort of mathematical fiat) the existence of a collection of objects (no matter which) where certain operations and relations are defined (no matter which) so that such and such hold – the such-and-such being purely formal (i.e. non-interpreted) expressions involving variables and constants for the objects, relations and operations stipulated, and maybe for higher-order entities also. In short, formal domains are the “abstract structures” of modern algebra. The concept appears, for instance, in the Prolegomena to Pure Logic, the first part of his Logical Investigations of 1900–1901.
- 15.
The use of group theory in the theory of algebraic equations is a perfect example of the methodological utility of formal equivalences in mathematics. The possibility of solving an equation by radicals is related to a formal property of a group associated with the equation. Galois connections establish ways of “translating” formal properties of a domain into properties of another domain.
References
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da Silva, J.J. (2011). On the Nature of Mathematical Knowledge. In: Krause, D., Videira, A. (eds) Brazilian Studies in Philosophy and History of Science. Boston Studies in the Philosophy of Science, vol 290. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9422-3_10
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